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Unformatted text preview: 614 Kinematics of Particles Fig. P11.9 and P11.10 Fig. P11.17 and P11.18 11.7 The motion of a particle is deﬁned by the relation x = 2t3 —
12t2  72t — 80, where x and t are expressed in meters and seconds, re
spectively. Determine (a) when the velocity is zero, (19) the velocity, the ac
celeration, and the total distance traveled when x = 0. 11.8 The motion of a particle is deﬁned by the relation x = 2t3 —
18t2 + 48t — 16, where x and t are expressed in millimeters and seconds, re
spectively. Determine (a) when the velocity is zero, (1)) the position and the
total distance traveled when the acceleration is zero. 11.9 The acceleration of point A is deﬁned by the relation (1 = —l.8
sin kt, where a and t are expressed in m/s2 and seconds, respectively, and
k = 3 rad/s. Knowing that x = 0 and v = 0.6 m/s when t = 0, determine the
velocity and position of point A when t = 0.5 s. 1 1 .10 The acceleration of point A is deﬁned by the relation (1 = — 1.08
sin kt — 1.44 cos kt, where a and t are expressed in m/s2 and seconds, re
spectively, and k = 3 rad/s. Knowing that x = 0.16 m and v = 0.36 m/s when
t = 0, determine the velocity and position of point A when t = 0.5 5. 11.11 The acceleration of a particle is directly proportional to the
square of the time t. When t = 0, the particle is at x = 36 ft. Knowing that
at t = 9 s, x = 144 ft and v = 27 ft/s, express 9: and o in terms of t. 11.12 It is known that from t = 2 s to t = 10 s the acceleraﬁon of a
particle is inversely proportional to the cube of the time t. When t = 2 s, v =
—15 ft/s, and when t = 10 s, v = 0.36 ft/s. Knowing that the particle is twice
as far from the origin when t = 2 s as it is when t = 10 s, determine (a) the
position of the particle when t = 2 s and when t = 10 s, (b) the total distance
traveled by the particle from t = 2 s to t = 10 s. 1 1. 13 The acceleration of a particle is directly proportional to the time
t. At t = 0, the velocity of the particle is 400 mm/s. Knowing that v =
370 mm/s and x = 500 mm when t = l 5, determine the velocity, the posi—
tion, and the total distance traveled when t = 7 s. 11.14 The acceleration of a particle is deﬁned by the relation a =
0.15 m/sz. Knowing that x = —10 m when t = 0 and v = —0.15 m/s when
t = 2 5, determine the velocity, the position, and the total distance traveled
when t = 5 5. 11.15 The acceleration of point A is deﬁned by the relation (1 =
200x(l + kxz), where a and x are expressed in m/s2 and meters, respectively,
and k is a constant. Knowing that the velocity of A is 2.5 m/s when x = 0 and
5 m/s when x = 0.15 m, determine the value of k. 11.16 The acceleration of point A is deﬁned by the relation (1 =
200x + 3200x3, where a and x are expressed in rn/s2 and meters, respectively.
Knowing that the velocity of A is 2.5 m/s and x = 0 when t = 0, determine
the velocity and position of A when t = 0.05 5. 11.17 Point A oscillates with an acceleration a = 2880 — 144x, where
a and x are expressed in in./s2 and inches, respectively. The magnitude of the
velocity is 11 in./s when x = 20.4 in. Determine (a) the maximum velocity of
A, (b) the two positions at which the velocity of A is zero. 11.18 Point A oscillates with an acceleration a = 144(20 — x), where
a and x are expressed in in./s2 and inches, respectively. Knowing that the sys
tem starts at time t = 0 with v = 0 and x = 19 in., determine the position
and the velocity of A when t = 0.2 s. re
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osition 11.19 The acceleration of a particle is deﬁned by the relation a =
1296 — 28, where a and x are expressed in m/s2 and meters, respectively.
Knowing that v = 8 m/s when x = 0, determine (a) the maximum value ofx,
(b) the velocity when the particle has traveled a total distance of 3 m. 11.20 The acceleration of a particle is deﬁned by the relation (1 =
k(l — e‘x), where k is a constant. Knowing that the velocity of the particle
is o = +9 m/s when x = —3 m and that the particle comes to rest at the
origin, determine (a) the value of k, (b) the velocity of the particle when
x = —2 m. 11.21 The acceleration of a particle is deﬁned by the relation (1 =
—k\/5, where k is a constant. Knowing that x = 0 and v = 25 ft/s at t = 0,
and that v = 12 ft/s when x = 6 ft, determine (a) the velocity of the particle
when x = 8 ft, (19) the time required for the particle to come to rest. 11.22 Starting from x = 0 with no initial velocity, a particle is given an
acceleration a = 08sz + 49, where a and v are expressed in ft/s2 and ft/s,
respectively. Determine (a) the position of the particle when u = 24 ft/s,
(b) the speed of the particle when x = 40 ft. 11.23 The acceleration of slider A is deﬁned by the relation (1 =
—2ka2 — 02, where a and v are expressed in m/s2 and m/s, respectively,
and k is a constant. The system starts at time t = 0 with x = 0.5 m and v =
0. Knowing that x = 0.4 in when t = 0.2 5, determine the value of k. 11.24 The acceleration of slider A is deﬁned by the relation (1 =
2 V 1 — 02, where a and v are expressed in m/s2 and m/s, respectively. The
system starts at time t = 0 with x = 0.5 m and v = 0. Determine (a) the
position ofA when u = —0.8 m/s, (1)) the position ofA when t = 0.2 5. 11.25 The acceleration of a particle is deﬁned by the relation (1 =
—kv2'5, where k is a constant. The particle starts at x = 0 with a velocity of
16 mm/s, and when x = 6 mm the velocity is observed to be 4 mm/s. De~
termine (a) the velocity of the particle when x = 5 mm, (b) the time at which
the velocity of the particle is 9 mm/s. 11.26 The acceleration of a particle is deﬁned by the relation (1 =
0.6(1 — k0), where k is a constant. Knowing that at t = 0 the particle starts
from rest at x = 6 m and that v = 6 m/s when t = 20 5, determine (a) the
constant k, (b) the position of the particle when u = 7.5 m/s, (0) the maxi
mum velocity of the particle. 11.27 Experimental data indicate that in a regiOn downstream of a
given louvered supply vent the velocity of the emitted air is deﬁned by v =
0.18vo/x, where v and x are expressed in ft/s and feet, respectively, and 00 is
the initial discharge velocity of the air. For 120 = 12 ft/s, determine (a) the
acceleration of the air at x = 6 ft, ([9) the time required for the air to ﬂow
fromx=3fttox= 10ft. 11.28 Based on observations, the s eed of a jogger can be approx—
imated by the relation 1) = 7.5(1 — 0.04x) '3, where v and x are expressed
in km/h and kilometers, respectively. Knowing that x = 0 at t = 0, deter—
mine (a) the distance the jogger has run when t = 1 h, (b) the jogger’s acceleration in m/52 at t = 0, (c) the time required for the jogger to run
6 km. Fig. P11.28 Problems 61 5 616 Kinematics of Particles Fig. P11.29 P ‘1"
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 Fig. P11.30 11.29 The acceleration due to gravity of a particle falling toward the
earth is a = —gR2/1‘2, where r is the distance from the center of the earth to
the particle, H is the radius of the earth, and g is the acceleration due to grav
ity at the surface of the earth. If R = 3960 mi, calculate the escape velocity,
that is, the minimum velocity with which a particle must be projected
upward from the surface of the earth if it is not to return to earth. (Hint:
o = 0 for r = <30.) 11.30 The acceleration due to gravity at an altitude y above the sur
face of the earth can be expressed as . A.
[1 + (y/20.9 x 106)]2 d— where a and y are expressed in ft/s2 and feet, respectively. Using this
expression, compute the height reached by a projectile ﬁred vertically upward
from the surface of the earth if its initial velocity is (a) o = 2400 ft/s, (b) o =
4000 ft/S, (c) 1) = 40,000 ft/S. 1 1 .31 The velocity of a slider is deﬁned by the relation 0 = o’ sin(wnt +
4)). Denoting the velocity and the position of the slider at t = 0 by 00 and x0,
respectively, and knowing that the maximum displacement of the slider is
2x0, show that (a) o’ = (00 + xgwﬁVZxown, (b) the maximum value of the ve
locity occurs when x = x0[3  (co/xown)2] / 2. 1 1 .32 The velocity of a particle is o = oo[1 — sin(1Tt/T)]. Knowing that
the particle starts from the origin with an initial velocity 00, determine (a) its
position and its acceleration at t = 3T, (22) its average velocity during the
intervalt=0tot =T. ________—_————— 11.4. UNIFORM RECTILINEAR MOTION Uniform rectilinear motion is a type of straight—line motion which is
frequently encountered in practical applications. In this motion, the
acceleration a of the particle is zero for every value of t. The veloc
ity o is therefore constant, and Eq. (11.1) becomes dx — = v = constant dt The position coordinate x is obtained by integrating this equation. De—
noting by x0 the initial value of x, we write fdx=vfotdt x—x0=vt x = x0 + vt (ll5) This equation can be used only if the velocity of the particle is known
to be constant. 11.37 A group of students launches a model rocket in the vertical di
rection. Based on tracking data, they determine that the altitude of the rocket
was 27.5 m at the end of the powered portion of the ﬂight and that the rocket
landed 16 5 later. Knowing that the descent parachute failed to deploy so that
the rocket fell freely to the ground after reaching its maximum altitude and
assuming that g = 9.81 m/sz, determine (a) the speed 01 of the rocket at the
end of powered ﬂight, (19) the maximum altitude reached by the rocket. 11.38 A sprinter in a 400—m race accelerates uniformly for the ﬁrst
130 m and then runs with constant velocity. If the sprinter’s time for the ﬁrst
130 m is 25 5, determine (a) his acceleration, (19) his final velocity, (0) his time
for the race. 11.39 In a close harness race, horse 2 passes horse 1 at point A, where
the two velocities are 02 = 7 m/s and 01 = 6.8 m/s. Horse 1 later passes horse
2 at point B and goes on to win the race at point C, 400 m from A. The
elapsed times from A to C for horse 1 and horse 2 are t1 = 61.5 s and t2 =
62.0 s, respectively. Assuming uniform accelerations for both horses between
A and C, determine (a) the distance from A to B, (b) the position of horse 1
relative to horse 2 when horse 1 reaches the ﬁnish line C . Fig. P1139 11.40 Two rockets are launched at a fireworks performance. Rocket
A is launched with an initial velocity 00 and rocket B is launched 4 5 later
with the same initial velocity. The two rockets are timed to explode simulta
neously at a height of 80 m, as A is falling and B is rising. Assuming a con
stant acceleration g = 9.81 m/s2, determine (a) the initial velocity 00, (b) the
velocity of B relative to A at the time of the explosion. 4..
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 Fig. P11.40 Problems 625 j Fig. P11.37 626 Kinematics of Particles Fig. P11.42 11.41 A police ofﬁcer in a patrol car parked in a 65 mi/h speed zone
observes a passing automobile traveling at a slow, constant speed. Believing
that the driver of the automobile might be intoxicated, the ofﬁcer starts his
car, accelerates uniformly to 85 mi/h in 8 s, and, maintaining a constant ve
locity of 85 mi/h, overtakes the motorist 42 s after the automobile passed
him. Knowing that 18 s elapsed before the ofﬁcer began pursuing the mo
torist, determine (a) the distance the ofﬁcer traveled before overtaking the
motorist, (b) the motorist’s speed. 11.42 As relay runner A enters the 65ftlong exchange zone with a
speed of 42 ft/s, he begins to slow down. He hands the baton to runner B
1.82 5 later as they leave the exchange zone with the same velocity. Deter
mine (a) the uniform acceleration of each of the runners, (b) when runner
B should begin to run. 11.43 In a boat race, boat A is leading boat B by 38 m and both boats
are traveling at a constant speed of 168 km/h. At t = 0, the boats accelerate
at constant rates. Knowing that when B passes A, t = 8 s and 0A = 228 km/h,
determine (a) the acceleration of A, (b) the acceleration of B. 11.44 Car A is parked along the northbound lane of a highway, and
car B is traveling in the southbound lane at a constant speed of 96 km/h. At
t = 0, A starts and accelerates at a constant rate aA, while at t = 5 s, B be
gins to slow down with a constant deceleration of magnitude aA/ 6. Knowing
that when the cars pass each other x = 90 m and 0A = 03, determine (a) the
acceleration (1A, (1)) when the vehicles pass each other, (0) the distance be
tween the vehicles at t = 0. (ugh, = 96 km/h
* B
A’UA Fig. P11.44 11.45 Two automobiles A and B traveling in the same direction in ad
jacent lanes are stopped at a trafﬁc signal. As the signal turns green, automo
bile A accelerates at a constant rate of 6.5 ft/sz. Two seconds later, automobile
B starts and accelerates at a constant rate of 11.7 ft/sz. Determine (a) when
and where B will overtake A, (b) the speed of each automobile at that time. 11.46 Two automobiles A and B are approaching each other in adja
cent highway lanes. At t = 0, A and B are 0.62 mi apart, their speeds are
DA = 68 mi/h and 03 = 39 mi/h, and they are at points P and Q, respectively.
Knowing that A passes point Q 40 s after B was there and that B passes point I
P 42 s after A was there, determine (a) the uniform accelerations of A and
B, {b} when the vehicles pass each other, (c) the speed of B at that time. 0A = 68 mi/h Fig. P11.46 628 Kinematics of Particles Fig. P11.55 and H156 11.51 In the position shown, collar B moves to the left with a constant
velocity of 300 mm/s. Determine (a) the velocity of collar A, (b) the velocity
of portion C of the cable, (0) the relative velocity of portion C of the cable
with respect to collar B. Fig. P11.51 and P11.52 11.52 Collar A starts from rest and moves to the right with a constant
acceleration. Knowing that after 8 s the relative velocity of collar B with re
spect to collar A is 610 mm/s, determine (a) the accelerations of A and B,
(b) the velocity and the change in position of B after 6 s. 1 1.53 At the instant shown, slider block B is moving to the right with
a constant acceleration, and its speed is 6 in./s. Knowing that after slider block
A has moved 10 in. to the right its velocity is 2.4 in./s, determine (a) the ac
celerations of A and B, (b) the acceleration of portion D of the cable, (0) the
velocity and the change in position of slider block B after 4 s. Fig. P11.53 and P11.54 11.54 Slider block B moves to the right with a constant velocity of
12 in./s. Determine (a) the velocity of slider block A, (b) the velocity of por
tion C of the cable, (a) the velocity of portion D of the cable, (d) the rela
tive velocity of portion C of the cable with respect to slider block A. 11.55 Collars A and B start from rest, and collar A moves upward with
an acceleration of St2 mm/sz. Knowing that collar B moves downward with
a constant acceleration and that its velocity is 200 mm/s after moving 800 mm,
determine (a) the acceleration of block C, (b) the distance through which
block C will have moved after 3 5. 11.56 Collar A starts from rest at t = 0 and moves downward with a
constant acceleration of 180 mm/sz. Collar B moves upward with a constant
acceleration, and its initial velocity is 200 Inm/s. Knowing that collar B moves
through 500 mm between t = 0 and t = 2 5, determine (a) the accelerations
of collar B and block C, (b) the time at which the velocity of block C is zero,
(0) the distance through which block C will have moved at that time. ...
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